Question

Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on
the remaining part of the circle.

Answer 1

Let the centre be O and the segment be AB.
Now join A to O and B to O.Angle AOB is a angle at centre .Let any point C on any part of the circle.Angle ACB will be any angle at the remaining part of the circle.
Now we have to prove:angle AOB=2angle ACB
We will draw aline passing through the centre from angle ACB.It will touch AB at point D.
angle ACO+angle OAC=angle AOD
Since AO=CO(radius)
Therefore, angleACO=angleOAC
2angle ACO=angleAOD........................(i)
angle BCO+angle OBC=angle BOD
Since BO=CO(radius)
Therefore, angleBCO=angleOBC
2angle BCO=angle BOD........................(ii)
Adding (i)and(ii)
2angleBCO+2angleACO=angle BOD+angle AOD
Therefore,angle AOB=2angleACB(BCO+ACO=ACB)